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05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z. 1 MODELAREA PROCESELOR FIZICO-CHIMICE C5, Miercuri, 04.11.2009, 12.00h, anii I (A+ C)

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  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...1

    MODELAREA PROCESELOR FIZICO-CHIMICE

    C5, Miercuri, 04.11.2009, 12.00h, anii I (A+ C)

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...2

    INSTRUMENTE MATEMATICE NECESARE

    (C4: 28-10-2009)

    1. Ecuatii diferentiale ordinare (EDO) 2. Ecuatii diferentiale cu derivate partiale(EDP) 3. Ecuatii cu diferente (EDt) 4. Transformata Fourier

    (C5: 04-11-2009)

    5. Transformata Laplace 6. Transformata z 7. Vectori si matrici 8. Algebra lineara 9. Variabile complexe

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...3

    Transformate, generalitati: Fourier, Laplace, z baze - Bibliografie[1]Jury,E.I.,Analysisandsynthesisofsampleddatacontrolsystems,

    CommunicationsandElectronics,1954,115[2]Lathi,B.P.,LinearSystemsandSignals,OUP,Oxford,NY,USA,2002[3]Levine,W.S.,TheControlHandbook,CRCPress,BocaRaton,Florida,

    USA,1996[4]Smith,S.W.,TheScientistandEngineersGuidetoDigitalSignal

    Processing,2ndedition,SanDiego:CaliforniaTechnicalPublishing,1999[5]Stanasila,O.,Analizamatematicaasemnalelorsiundinelor,MatrixRom,

    Bucuresti,1997[6]Duhamel,P.andM.Vetterli,"FastFourierTransforms:ATutorialReview

    andaStateoftheArt,"SignalProcessing,Vol.19,April1990,pp.259299.[7]FFTW(http://www.fftw.org)[8]Ingle,V.K.andJ.K.Proakis,DigitalSignalProcessing.UsingMatlabV.4,

    PWSPublish.Co.,ITP,Boston,MA,USA,1997[9]Levine,W.S.,TheControlHandbook,CRCPress,BocaRaton,Florida,

    USA,1996[10]http://en.wikipedia.org/wiki/Fourier_Laplace_Ztransform[11]http://www.fftw.org

    http://en.wikipedia.org/wiki/Fourier_Laplace_Z-transformhttp://www.fftw.org/http://www.fftw.org/

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...4

    Transformata Fourier Proprietati

    TFareoseriedeproprietati,cuajutorulcarorasepotdeterminaTFalemultorfunctiicomune:

    Linearitate TFafunctiilordeplasateintimplastangasauladreapta TFafunctiilorscalateintimp Inversarea(semnului)timpului Multiplicareacuoputereatimpuluit Multiplicareacuexponentialafrecventeiexp(j0t) Multiplicareacusinusulfrecventeisin(0t) Multiplicareacucosinusulfrecventeicos(0t) TFaderivateiuneifunctiiindomeniultimp Inmultireaindomeniultimp Convolutiaindomeniultimp Proprietateadedualitate TeoremaluiParsevals.a.

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...5

    TRANSFORMATA LAPLACE (TL)

    Inanalizafunctionala,TLauneifunctiioriginalf(t)definitapentrut0,estefunctiaimagineF(s),definitade(TLdirecta):

    Undes=+iestevariabilacomplexas,iarlimitainferioaraaintegralei,adica0estenotatiacarereprezinta

    siasiguraincluziuneaintregiifunctiiDiracdelta,adica(t),in0(origine),dacaacoloesteunastfeldeimpulsinf(t),la0.

    Observatie;TLestenumitaastfel,inonoarealuiPSLaplace(autilizatoinlucrareasadespreteoriaprobabilitatilor),darafostdescoperitadeLeonhardEuler.

    TLexistapentrutoatenumerelecomplexecareauRe{s}>c,,undecesteoconstantareala,numitaabscisadeconvergenta.LaTLbilaterala,decilimiteledeintegrareintresi+,existentaacesteia(adicaundeesteeadefinita),esteintrelimitelecd.Adica,valorileluispentrucareexistaTL,senumesteregiunedeconvergenta(RDC),saudomeniudeconvergenta(DDC).

    TLinversaesteointegralacomplexatipBromwichWagner:

    Undeesteunnumarreala.i.conturuldeintegrareesteininteriorulRDCaluiF(s),

    iari=sqrt(1).

    TLbilateralaestedefinitaderelatia:

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...6

    TRANSFORMATA LAPLACE (TL)Legatura cu transformata Fourier

    OBSERVATIE:TFcontinua(lectiatrecuta)esteechivalentacuTLbilaterala,ceareargumentcomplex,s=i:

    Relatianuarefactoruldescalare1/sqrt(2*)=()delaTF,darseutilizeazaladeterminareaspectruluidefrecventaalunuisemnalaferentunuisistemdinamic.

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...7

    TRANSFORMATA LAPLACE (TL)Proprietati si teoreme

    CasiTF,TLareoseriedeproprietatisiteoreme,cuajutorulcarorasepotdeterminaTLdirectesauinversealemultorfunctiicomune:

    Linearitate TLafunctiilordeplasateintimpladreapta(intarziere,retardare) TLafunctiilorcutimpscalat Multiplicareacuoputereatimpuluit Multiplicareacusinusulfrecventeisin(0t) Multiplicareacucosinusulfrecventeicos(0t) TLaderivateiuneifunctiiindomeniultimp TLaintegraleiuneifunctiiindomeniultimp Convolutiaindomeniultimp Teoremavalorii(limita)initiale Teoremavalorii(limita)finale

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...8

    TRANSFORMATA LAPLACE (TL)Proprietati si teoreme (I)

    Linearitate

    Derivare(teoremaderivarii)

    Teoremaintegrariiinfrecventa:

    Teoremaintegrariiintimp:

    Teoremavaloriiinitiale:

    Teoremavaloriifinale:

    totipoliifiindinSSPD;TVFaratacomportareainregimstationar(SS),faraafacedescompunereainfractiipartialesimpleetc.

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...9

    TRANSFORMATA LAPLACE (TL)Proprietati si teoreme (II)

    Teoremadeplasariiinfrecventa:

    Teoremadeplasariiintimp:

    Teoremaconvolutiei:

    Multiplicarecuoputereatimpuluit:

    undeDestesimbolulderivatei[aici,deordinuln,afunctieiF(s)]

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...10

    TRANSFORMATE LAPLACErationale(i)

    TLauneifunctii,X(s),seziceesteofunctierationalades,dacapoatefiscrisacaunraportdepolinoameins,ireductibil,adica

    X(s)=N(s)/D(s),UndeN(s)siD(s)suntpolinoameinvariabilacomplexas,datede:

    N(s)=bmsm+bm1sm1+.+b1s+b0D(s)=sn+an1sn1+.+a1s+a0,

    puterilemsinfiindintregipozitivi,iarcoeficientiibm,bm1,.,b1,b0sian1,an2,,a1,a0suntnumerereale.

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...11

    TRANSFORMATE LAPLACErationale(ii-continuare)

    PolinomulD(s)sezicecaestemonic,dacaarecoeficientulan,alluisn,egalcu1.

    Intreguln,gradulluiD(s),senumesteordinulfunctieirationaleX(s)

    Dacanm,X(s)sezicecaesteofunctierationalaproprie.

    Dacan>m,X(s)sezicecaesteofunctierationalastrictproprie

    RadacinilepolinomuluiD(s)(ecuatiaD(s)=0),senumescpoliilui X(s)

    RadacinilepolinomuluiN(s)(ecuatiaN(s)=0),senumesczerourileluiX(s)

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...12

    TRANSFORMATE LAPLACEirationale

    DacaTLX(s)auneifunctiix(t)nuesteofunctierationala,sezicecaesteirationala.Adica,X(s)nupoatefiexprimatacaunraportdepolinoameins.

    Exemplu:X(s)=et0s/sesteirationala:exponentialanupoatefifacutaunraportdepolinoame

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...13

    TRANSFORMATA Z (TZ)generalitati

    TransformataZ,inmatematica,automatica,modelare,procesareasemnaleloretc.,convertesteunsemnaldindomeniultimpdiscret(deciosecventadenumerereale),introreprezentareindomeniulfrecventacomplexa.

    Observatii: TZafostintrodusa,cuacestnume,decatreE.I.Juryin1958,in

    SampledDataControlSystems,edituraJohnWiley&Sons. AsacumtransformataLaplaceartrebuisasenumeasca

    transformatas,transformataZartebuisasenumeascatransformataLaurent,intrucataceastasebazeazapeseriaLaurent.

    TZunilateralaestepentrusemnaleleintimpdiscret,ceeaceesteTLunilateralapentrusemnaleleintimpcontinuu.

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...14

    TRANSFORMATA Z directa

    TZaunuisemnalintimpdiscretz[n]este,prindefinitie,functiaimagineX(z),data(definita)derelatia(numitasiTZdirecta):

    Undenesteunintreg,iarzesteunnumarcircularcomplexdeformaz=rej.

    TZdemaisus,culimitelesumeidela+la,semainumestesiTZbilaterala,TZpeambelefete,TZinfinitdubla

    AtuncicandlimitelesumeiserieiLaurentsuntdelazero(0)la+,seziceTZunilaterala,sauTZpeosingurafata(utilizatalasemnalecauzale,adicaceledefinitepentrut0):

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...15

    TRANSFORMATA Z inversa

    TZinversaestedataderelatiademaijos,undeCesteunconturinchisantiorarceincercuiesteoriginea,intreagaregiunedeconvergenta(RDC)sitotipoliiluiX(z).

    UncazspecialalacesteiintegraledeconturesteatuncicandconturulCestecercunitate(discunitateadicaarerazaegalacu1),RDCincludecerculunitate,iarintegralademaisusdevinetransformataFourierinversaintimpdiscret:

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...16

    TRANSFORMATE Z rationale

    CasilaTL,TZpoatefiofunctierationalainz,adica:X(z)=N(z)/D(z),undeN(z)siD(z)suntpolinoamein

    variabilacomplexaz,datede:

    N(z)=bmzm+bm1zm1+..+b1z+b0,D(z)=zn+an1zn1+.+a1z+a0

    DacasepresupunecaX(z)esteproprie,atunciavemnm. X(z)strictproprie,apoipoliisizerourilefunctieirationaleX(z),au

    acelasiintelescasilaTL. DacaX(z)estesubformarationala,transformatazinversax[n]se

    poatecalculadezvoltandpeX(z)introseriedeputeriinz1,impartindpeN(z)laD(z)prinimpartireinfinita(numitauneorisiimpartiresintetica,impartirelunga),valorilex[n]fiindcoeficientiiluizn

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...17

    CALCULAREA T.F., T.L. SI T-Z IN MATLAB(i)

    InSymbolicMathToolbox(TLdirectasiinversa): laplacecalculeazatransformataLaplacedirectaauneifunctiiintsirezultao

    functieins(cf.relatieidedefinitie); Exemplul1:

    >>symst%aratacaoricevariabilaindicatavafi %utilizatasimbolicsinucaovaloarenumerica

    %aici,f(t)=t(functiaoriginal)

    >>f=tf=t>>laplace(f)ans=1/s^2

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...18

    CALCULAREA T.F., T.L. SI T-Z IN MATLAB(ii)

    InSymbolicMathToolbox(TLdirectasiinversa):

    ilaplacecalculeazatransformatainversaLaplaceauneifunctiiinssirezultaofunctieint(cf.relatieidedefinitie);

    Exemplul2:

    >>symss%aratacaoricevariabilaindicatavafi %utilizatasimbolicsinucaovaloarenumerica

    >>F=1/(s^2)F=1/s^2>>ilaplace(F)ans=t

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...19

    CALCULAREA T.F., T.L. SI T-Z IN MATLAB(iii)

    InSymbolicMathToolbox(TFdirectasiinversacontinua): fourier(x)- calculeazatransformataFourierdirectaaunei

    functiiintimpcontinuudex(variabilasimbolica)sireturneazaofunctieinw(w=omega,deci,conformdefinitieiavem:f=f(x) F=F(w));

    dacaf=f(w),fourierreturneazaofunctiedet:F=F(t).

    ifouriercalculeazatransformatainversaFourierauneifunctiiinwsirezultaofunctieinx(cf.relatieidedefinitie:F=F(w) f=f(x)).Aici,xesteovariabilasimbolica,lafelt=timp;

    dacafunctiaF=F(x),ifourierreturneazaofunctiedet,adica:f=f(t)

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...20

    CALCULAREA T.F., T.L. SI T-Z IN MATLAB(iv)

    InMatlab(TransformataFourierDiscretaTFD):

    FunctiileX=fft(x)six=ifft(X)implementeazaperecheadetransformateFourierdiscretedirectasiinversadatepentruvectoriidelungimenderelatiile(vezisicursultrecut,undeexp(2i/n)esteanaradacinaaunitatii):

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...21

    CALCULAREA T.F., T.L. SI T-Z IN MATLAB(v)

    InSymbolicMathToolbox(Transformatazdirectasiinversa): ztransestetransformatazdirectaavariabileiscalaresimboliceintimp

    discret(devariabilaindependentan),infunctiaimaginedez,F(z):f=f(n) F=F(z),conformcucomandaF=ztrans(f),respectivrelatia:

    F(z)=f(n)/zn=f(n)*zn

    Exemplul3:>>symsn>>F=2^n*nF=2^n*n>>ztrans(F)ans=1/2*z/(1/2*z1)^2

    0

    0

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...22

    CALCULAREA T.F., T.L. SI T-Z CU MATLAB (IV)- (Tr-Z)

    %Examplul4.Sedasirul/secventax(n)=anu(n),0x='a^n';>>X=ztrans(x)_______________________________________

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...23

    CALCULAREA T.F., T.L. SI T-Z IN MATLAB(vi)

    InSymbolicMathToolbox(Transformatazinversa):- iztransestetransformatazinversa,adicaf=iztrans(F)a

    obiectuluisimbolicscalarFdevariabilaindependentaz.Functiareturnataesteunadevariabilaindependentan,adica,f = iztrans(F)

    Exemplul5:>>symsz>>f=2*z/(z2)^2f=2*z/(z2)^2>>iztrans(f)ans=2^n*n

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...24

    CALCULAREA T.F., T.L. SI T-Z CU MATLAB (II)

    Exemplul6(Matlabhelp).Sasegaseascacomponenteledefrecventaaleunuisemnalinnecat(amestecat)cuunsemnalzgomot,dindomeniultimp.Seconsideracadatelesuntesantionatela1000Hz.Semnalulcontinecomponentede50Hzsi120Hzsiesteamestecatcuzgomotaleatordemediezero.

    >> zgomot aleator de medie zero: t = 0:0.001:0.6;>> x = sin(2*pi*50*t)+sin(2*pi*120*t);>> y = x + 2*randn(size(t));>> plot(1000*t(1:50),y(1:50))>> title('Semnal amestecat cu zgomot aleator de medie zero)>> xlabel('timp (milisecunde)')_________________________________________________________>> frequency axis: f = 1000*(0:256)/512;>> plot(f,Pyy(1:257))>> title(Continutul de frecvente a lui y')>> xlabel(Frecventa (Hz)')

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...25

    CALCULAREA T.F., T.L. SI T-Z CU MATLAB (IIIc)-implementare Matlab

    Exemplul7.Calcululexactsiaproximativalspectrului|F()|(TFadica),6pentrufunctiaimpulsdeordinul2,datade:f(t)=1pentru0t2sif(t)=0,inrest(IIIb).

    >>N=128;%AproximareaemaibunadacasemaresteN(256,512)si/sausemicsoreazaT

    >>T=0.1;>>N=input(InputN:);>>T=input(InputT:);>>t=0:T:2;>>f=[ones(1,length(t))zeros(1,Nlength(t))]>>Fn=fft(f);>>gam=2*pi/N/T;>>n=0:10/gam;>>Fapp=(1exp(j*n*gam*T))/j/n/gam*Fn;>>w=0:.05:10;>>Fexact=2*sin(w)./ww;>>plot(n*gam,abs(Fapp(1:length(n))),og,w,abs(Fexact),b)

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...26

    CALCULAREA T.F., T.L. SI T-Z CU MATLAB (II) (de refacut)REZERVE

    Exemplele8.13,celesasedatepentrulaborator(laplace,ilaplace,ztrans,iztrans,fourier,ifourier),careutilizeazacalcululsimboliclacalculareatransformatelorFourier,Laplacesiz

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...27

    MATRICI

    MATRICI Algebramatriceala Inversareauneimatrice Determinant,determinantisimatriciinverse Matricitranspuse Matricibloc Puteridematrici,polinoamematriceale

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...28

    ALGEBRA LINEARA(I)

    Spatiivectoriale Definitii,exemple,proprietati Functiilineare Normediverse Produsscalar

    Ecuatiilineare(EL) Unicitateasiexistentasolutiilor SolutiaEL Aproximareasolutiilor Pseudoinversauneimatrice

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...29

    ALGEBRA LINEARA(II)

    Valoripropriisivectoriproprii Definitii,proprietati Bazecuvectoriproprii,diagonalizare Valoriproprii,proprietati

    Forma(canonica)Jordan,similaritateamatricelor Descompunereainvalorisingulare(DVS) AlgebralinearasiMatlab

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...30

    VARIABILE COMPLEXE Numerecomplexe(NC)(algebra,conjugarea,modululsi

    reprezentaregeometricaaNC); Functiicomplexe(ecuatiidiferentialeCauchyRiemann,

    polisizerouri,functiirationale(dezvoltareainfractiipartialesimple,formulaluiLucas,functiimeromorfe),dezvoltareafunctiilorinseriideputeri)

    Integralecomplexe(teoremaluiCauchy,principiulargumentului(criteriuldestabilitateNyquist),teoremareziduurilor).

    Aplicatii/reprezentari/mapari(=mappings)conforme(transformaribilineare(sautransformarifractionarelineare),exemple)

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...31

    ULTIMUL SLIDE: TC3 pentru data viitoare

    Temadecasa5(TC5):

    SaseexecuteutilizandMatlab,toateexempleledemaisus(13),datepentrutransformateleFourier,Laplacesiz,utilizandatatcalcululnumericcatsicalcululsimbolic)

    Multumesc,saptamanaviitoare!

  • 05.11.09 MPFC, C5_Instrumente de modelare II (continuare): transformatele Laplace, z...32

    Recapitulare titluri teme de casa (TC):

    TC1:5pagcu10functiidiferiteMatlab,laalegere;cateofunctiepepagina(laimprimanta!),dar,chestiunidiferitecuACEEASIfunctie

    TC2:6pagcu6functiisimbolicediferite,laalegere,cateofunctiepepagina(laimprimanta!),dar,chestiunidiferitecuACEEASIfunctie.

    TC3:10pagcufunctiilefft, ifft(tr.Fourierauneisecvente),sifunctiilebartlett; blackman; blackmanharis; hann; chebwin;kaiser; tukeywin; rectwin; gausswin; bohmanwin (TFdirectasiinversacuferestrediversepetimpscurt)

    TC4:Modificarea,adaugarea,revizuireafunctieiwhydinMATLAB,pentruasepotrivigusturilordvs.proprii(elevajunsstudent:glumet;serios;neserios;cercetator;informatician;automatist;meticulos;bigot;amic/fiu/fiicabun/rau;visator;romantic;educat;cult;civilizatetc)

    TC5:Cele13exempledemaisus,visvisdetransformateleLaplace,Fouriersiz

    MODELAREA PROCESELOR FIZICO-CHIMICEINSTRUMENTE MATEMATICE NECESARETransformate, generalitati: Fourier, Laplace, z baze - BibliografieTransformata Fourier Proprietati TRANSFORMATA LAPLACE (TL)TRANSFORMATA LAPLACE (TL) Legatura cu transformata FourierTRANSFORMATA LAPLACE (TL) Proprietati si teoremeTRANSFORMATA LAPLACE (TL) Proprietati si teoreme (I)TRANSFORMATA LAPLACE (TL) Proprietati si teoreme (II)TRANSFORMATE LAPLACE rationale(i)TRANSFORMATE LAPLACE rationale(ii-continuare)TRANSFORMATE LAPLACE irationaleTRANSFORMATA Z (TZ) generalitatiTRANSFORMATA Z directaTRANSFORMATA Z inversaTRANSFORMATE Z rationaleCALCULAREA T.F., T.L. SI T-Z IN MATLAB(i)CALCULAREA T.F., T.L. SI T-Z IN MATLAB(ii)CALCULAREA T.F., T.L. SI T-Z IN MATLAB(iii)CALCULAREA T.F., T.L. SI T-Z IN MATLAB(iv)CALCULAREA T.F., T.L. SI T-Z IN MATLAB(v)CALCULAREA T.F., T.L. SI T-Z CU MATLAB (IV)- (Tr-Z)CALCULAREA T.F., T.L. SI T-Z IN MATLAB(vi)CALCULAREA T.F., T.L. SI T-Z CU MATLAB (II)CALCULAREA T.F., T.L. SI T-Z CU MATLAB (IIIc)-implementare Matlab CALCULAREA T.F., T.L. SI T-Z CU MATLAB (II) (de refacut)REZERVEMATRICIALGEBRA LINEARA(I)ALGEBRA LINEARA(II)VARIABILE COMPLEXE ULTIMUL SLIDE: TC3 pentru data viitoareRecapitulare titluri teme de casa (TC):